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Depolarization of light scattered by a single sodium nanoparticle trapped in an electro-optical trap
1 October 2002
Optics Communications 211 (2002) 171–181 www.elsevier.com/locate/optcom
Depolarization of light scattered by a single sodium nanoparticle trapped in an electro-optical trap W. Bazhan *, K. Kolwas, M. Kolwas Institute of Physics of Polish Academy of Sciences, Al. Lotnik ow 32/46, 02-668 Warsaw, Poland Received 8 April 2002; received in revised form 21 June 2002; accepted 7 August 2002
Abstract Scattering of polarized light by a single metal particle is studied. Single sodium particles trapped in an electro-optical trap show the unexpectedly large cross-polarized scattered intensities not predicted by Mie theory. To explain qualitatively the large depolarization effect, spheroidal shape of the particle is assumed and the T-matrix method (TMM) is applied to study the corresponding scattered intensities and the resulting depolarization rates. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 41.20.)q; 42.25.Ja; 42.68.Mj; 77.22.Ej Keywords: Particle trapping; Scattering of light by nonspherical objects; T-matrix method
1. Introduction Quantitative interpretation of light scattering by small particles (aerosols, clouds, colloidal particles, large metal clusters and so on) requires accurate knowledge of the interaction of small objects with electromagnetic radiation. Although many natural particles are known to have nonspherical shapes, the relative simplicity and easy applicability of Mie theory [1] has led to widespread practice of treating scattering particles as spheres. In fact, a vast amount of accumulated evidence suggests that scattering properties of
*
Corresponding author. Fax: +48-22-843-0926. E-mail address: [email protected] (W. Bazhan).
nonspherical particles can differ both quantitatively and qualitatively from those of ÔequivalentÕ spheres. One of the most popular modern techniques of modeling the light scattering by nonspherical objects is the T-matrix method (TMM) firstly introduced by Waterman [2] and significantly improved by Mishchenko [3]. Most papers published during the last 20 years, employed the TMM for interpreting the light scattering by randomly oriented axially symmetric mono- or polydisperse nonabsorbing particles. But very few of them were dedicated to applying the T-matrix techniques to the case of light scattering by single, free particles and especially by single free metal particles. Such tiny objects have rarely been examined under laboratory conditions, what has mainly been caused by difficulties in fabrication
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 9 9 - 0
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and localization for investigation. Currently the most developed nano- and microparticle localizing techniques is particle trapping. There are many reports concerning various combinations of particle trap construction (e.g. [4,5]) and trapping conditions. But the area of metallic particle trapping in conducting gaseous media is still virtually unexplored since it presents some substantial technical difficulties [6,7]. In this paper we report the results of experimental investigation of scattering of linearly polarized laser light by single trapped sodium clusters in a cold plasma environment. We used the TMM tools for interpreting the unexpectedly large depolarization of the scattered light observed in our experiments.
2. Experimental setup Sodium clusters are produced in a heat-pipe type stainless-steel cell [8] with crossed tube design [9,10]. The inner part of the cell is heated by an electrical oven, which ensures a uniform central temperature region. In standard working conditions used in the present experiments, the temperature is stabilized at 650 K. This temperature ensures an appropriate sodium vapor density, essential for a sufficient amount of Na2 molecules (dimers) over the sodium deposit placed in the central part of the cell (see [10]). The presence of dimers enables to supersaturate the vapor by light, the process described in detail in earlier papers [9,10]. The arms of the cell are cooled with flowing water in order to keep glass cell windows at a safe temperature. To prevent sodium deposition on the cooled windows of the cell, we buffer it with helium gas. Due to the heat-pipe features of the cell [8], sodium contaminants which could not be avoided during preparation of the sodium deposit, are pushed out from the central region of the cell towards colder arms, owing to the high sodium vapor pressure in the central high temperature region. Helium buffer gas, which is chemically neutral with respect to sodium, plays two further important roles due to its high pressure (in the present experiment typically about 640 Torr). In the
process of collisions with dimers, helium atoms effectively assist the process of dissociation of dimers previously excited by laser light. In addition, the helium gas serves as a reservoir of heat released during cluster formation, enabling effective spontaneous growth of clusters in supersaturated sodium vapor. Two types of clusters are produced in the cell. The first type are Ôlight inducedÕ in the central, hot region of the cell. The hot clusters, sodium droplets, appear only in the presence of the laser light with frequency in resonance with the well-defined sodium dimer transition enabling excitation of an excess of atoms in previously saturated vapor, constituting the source of supersaturation and enabling the spontaneous growth of sodium droplets from atomic size scale up to large sodium droplets of about 300 nm in diameter (see [10] for details). The process of sodium cluster growth lasts from few to few tens of seconds after laser light with sufficiently large power (100–400 mW) is switched on. Clusters of a second type are produced in the arms of the cell, in the gradient temperature zones near the windows. Sodium vapor reaching that region is cooled down and thus supersaturated, becoming the source of the large sodium clusters growing on contaminations (oxides and hydrates of sodium) collected in those zones. Such contaminants constitute condensation embryos leading to clusters whose shapes are expected to differ in an important manner from the spherical sodium droplets induced by light. Even at relatively small oven temperatures, and low laser light power, we can see a lot of particles localized in cold parts of the cell arms. To investigate a single particle by optical means we use an electro-optical trap described in more detail in [6,7]. Its electric part consists of the quadrupole [4] allowing for trapping in two dimensions, adapted for working in a hot dense gaseous medium. The quadrupole consists of four (cylindrical) metal rods which are 1 mm apart and are electrically insulated with fused-silica tubes (see Fig. 1). This insulation is essential for trapping particles the electrodes. On the other hand the presence of the conducting medium greatly reduces accumulation of charged clusters on the electrode
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
Fig. 1. Schematic diagram of the electro-optical trap.
insulation, so that there is less distortion in the shape of the trapping electric potential. The trapping in the third dimension is assured by the light gradient force [7] of the 488 nm line of the Arþ laser working at 400 mW. The trapping beam is directed perpendicular to the quadrupole axis. The trap is mounted in the central region of the cell. The experimental setup is sketched in Fig. 2. A detailed discussion of the trapping mechanism and the description of the papers [6,7]. Our previous experiments were directed to trap the sodium droplets generated by light. However, we noticed, that after the trap is switched on, the Ôcold armsÕ clusters were efficiently driven by the electrostatic forces into the trap, in which they could be kept for a long time (some tens of minutes). In contrast to the light induced sodium droplets, those large particles are expected to be of irregular shape. Scattering experiments performed
Fig. 2. Schematic diagram of the experimental setup.
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on those objects produced such rarely seen result (up to 100% of light depolarization in the direction of observation), that it itself became a subject of investigation. The scattering experiment on the trapped particle is performed with the additional (40 mW) probe light beam of the Argon–Krypton laser at wavelength of 647 nm (with the aperture of about 5 mm), directed perpendicular to the quadrupole axis and opposite to the Arþ trapping laser beam. In contrary to the Arþ laser beam, the probe beam is not focused. In such conditions, the stability of the particle inside the trap is not disturbed by the presence of the weak probe beam and the probe intensity which illuminates the particle is almost isotropic. The intensity of the elastically scattered light from the probe beam is observed at right angle to the direction of the probe beam, and after passing through a polarization analyzer, a color filter and a microscope system, is measured by a monochromatic CCD camera installed on the microscope eyepiece (see Fig. 2).
3. Experimental scattered intensities The intensity of elastically scattered probe beam light is measured as a function of time at right angle to the direction of the laser beam propagation. The experimental signal is gathered in four different polarization geometries: (i) The polarization of the incident light perpendicular to the observation plane, and the polarization of the observed light perpendicular or parallel to the observation plane (on the basis of the usual s–p convention, we denote this as ss and sp polarization geometry, respectively (see Fig. 3)). (ii) The polarization of the incident light parallel to the observation plane and the polarization of observed light parallel or perpendicular to the observation plane (pp and ps polarization geometry, respectively (see Fig. 3)). The positions of the polarizer and of the analyzer assuring respective sequential combinations: ss ! sp ! pp ! ps are controlled by a PC with appropriate measurement of the time intervals. The complete time of a single measurement is
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Fig. 3. Polarization geometries of the scattering experiment.
nearly 5 s and the time between two sequential measurements adds to a total of 10 s. Electronic control of the relative positions of polarizers, of the data acquisition and numerical processing of the scattered intensities in the CCD camera image allowed an improvement in the quality of results with respect to our previous trapping experiment [6,7]. Figs. 4 and 5 illustrate the measured intensities of the light scattered by a single cluster during its residence in the trap and the resulting depolarization coefficients dps and dsp : dps ¼ 100 Ips =Ipp ;
ð1Þ
dsp ¼ 100 Isp =Iss :
ð2Þ
Fig. 4. The intensities Iss , Ipp (ÔparallelÕ polarization geometries) and Ips , Isp (ÔcrossedÕ polarization geometries) of light scattered by a single trapped sodium particle, measured during the residence of the particle in the trap for kscat ¼ 647 nm, uac ¼ 338 V, x ¼ 13 190 s1 and udc ¼ 40 V.
Fig. 5. The depolarization coefficients dps and dsp for light scattered by a single trapped sodium particle, measured during the residence of particle in the trap for kscat ¼ 647 nm, uac ¼ 338 V, x ¼ 13 190 s1 and udc ¼ 40 V.
A single experimental measurement comprises of 16 serial images for the four sequential combinations of the polarizer–analyzer system: ss ! sp ! pp ! ps. The observed time dependencies of the intensities Iss and Ipp of the scattered fight for ss and pp polarizations, respectively (see Fig. 4), show that the trapped cluster changes its size with time. As illustrated in Figs. 4 and 5, we observed also relatively large intensities Ips and Isp for the ÔcrossedÕ ps and sp polarizations. Mie scattering theory does not predict non-zero intensity corresponding to the ÔcrossedÕ polarization geometries: the depolarized components Ips and Isp in the scattered light are absent. Mie theory is formulated for the electromagnetic wave scattered by a sphere with a smooth surface and sharp boundary and with a well-defined dielectric function, which is an adjustable parameter of the theory. It relies on solving MaxwellÕs equations together with the boundary conditions separated into a set of ordinary differential equations for the two subfields (internal and external) in the form of infinite series [1]. We interpret the presence of the cross-polarization intensity components in scattering experiments as being due to the fact that the investigated scatterers are not perfect spherical objects. If so, Mie theory cannot be used for adequate interpretation of the observed depolarization of the scattered light.
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The aim of the present paper is to check, whether the hypothesis concerning the nonsphericity of the shape of the scatterer is sufficient to interpret the presence of unexpectedly large crosspolarization intensities Ips and Isp of the ÔcrossedÕ ps and sp polarization geometries. For this purpose we used the formalism of the TMM for spheroidal particles as relatively the most simple shapes of scatterers one can assume for explaining the observations. The appropriate computer code created for that purpose results in qualitative agrement with experiment that cannot be achieved by means of Mie theory.
4. TMM scattering intensities The main purpose of this section is to recall the theoretical basis of the TMM and to use it for interpretation of the scattered light depolarization effects observed in our experiments. TMM, or Watermans extended boundary condition method [2] (EBCM), has shown its high potential for studying scattering by nonspherical particles. Although it can be applied to virtually any kind of nonspherical particles, it is most efficient for axisymmetric scatterers. A detailed review of the EBCM is given by Tsang et al. [11] and Mishchenko et al. [12]. In the T-matrix approach, the incident Einc and the scattered Esca fields are expanded into series of vector spherical harmonics Mmn and Nmn , respectively, as follows: inc
E ðRÞ ¼
1 X n X
½amn RgMmn ðkRÞ
n¼1 m¼n
þ bmn RgNmn ðkRÞ ; Esca ðRÞ ¼
1 X
n X
ð3Þ
½pmn Mmn ðkRÞ
n¼1 m¼n
þ qmn Nmn ðkRÞ ;
jRj > r0 ;
ð4Þ
where k ¼ 2p=k is the free-space wavenumber for free-space wavelength k; R is the radius vector with its origin at the origin of the coordinate system; r0 is the radius of a circumscribing sphere of the scattering particle; amn ; bmn and pmn ; qmn are incident field expansion coefficients and scattered field
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expansion coefficients appropriately. The analytical formulas for amn and bmn are introduced by Waterman [2]. The expressions for the functions RgMmn and RgNmn can be obtained from the functions Mmn and Nmn by replacing the spherical Hankel functions hnð1Þ in its definition by the spherical Bessel functions jn as it was shown by Mishchenko et al. [12]. In addition, the internal field is also expanded as follows: Eint ðRÞ ¼
1 X n X
½cmn RgMmn ðmr kRÞ
n¼1 m¼n
þ dmn RgNmn ðmr kRÞ ;
ð5Þ
where mr is the refractive index of the particle relative to that of the surrounding medium; cmn and qmn are internal field expansion coefficients. The relation between the scattered field coefficients pmn and qmn and the incident field coefficients amn and bmn is linear and is given by a transition matrix T [2,11]: pmn ¼
1 n0 X X
11 12 Tmnm 0 n0 am0 n0 þ Tmnm0 n0 bm0 n0 ;
ð6Þ
n0 ¼1 m0 ¼n
qmn ¼
1 n0 X X
21 22 Tmnm 0 n0 am0 n0 þ Tmnm0 n0 bm0 n0 :
ð7Þ
n0 ¼1 m0 ¼n
In the more compact matrix notation, Eqs. (6) and (7) can be written as 11 P a T T12 a ¼T ¼ : ð8Þ q b T21 T22 b Similarly, the relation between the exciting field coefficients and the internal field coefficients is 11 a Q12 c Q ; ð9Þ ¼ b Q21 Q22 d where the elements of the matrix Q are twodimensional integrals, which must be evaluated numerically over the particle surface, and depend on the particle size, shape, refractive index, and orientation with respect to the natural reference frame [11]. As the scattered field coefficients can also be expressed by the internal field coefficients p RgQ11 RgQ12 c ¼ ; ð10Þ q RgQ21 RgQ22 d
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The T matrix can be represented as 1
T ¼ RgQ½Q :
ð11Þ
The scattering properties of a particle are completely characterized by the elements of the amplitude scattering matrix that relates the scattered field to the the incident filed. The amplitude scattering matrix can be determining as sca inc expðikRÞ L sca inc Ep Ep S ðn ; n ; a; b; cÞ ¼ inc ; Esca R E s s ð12Þ L
where S is the 2 2 amplitude matrix that transforms the electric field vector components Epinc ; Esinc of the incident wave into the electric field vector components Epsca ; Essca of the scattered wave in the laboratory reference frame. The latter is a right-handed Cartesian coordinate system, with orientation fixed in space, having its origin inside the particle [12]. The direction of propagation of a transverse electromagnetic wave is specified by a unit vector n or, equivalently, by a pair (hL ; uL ) of polar and azimuthal angles (see Fig. 6). The amplitude matrix depends on the directions of the incident and of the observed scattered light as well as on the size, morphology, and the particle natural reference frame with respect to the laboratory
z n φ θ θ y φL
x Fig. 6. The laboratory coordinate system.
reference frame as specified by the Euler angles of rotation a, b, and c [13]. In order to calculate the SL amplitude matrix, the T matrix should first be found for a given scattering particle. It is rather convenient to specify the amplitude matrix with respect to the particle reference frame by using the T-matrix method, and then to obtain the SL amplitude matrix by means of the coordinate system transformation, as has been explicitly shown by Mishchenko [14]. By taking into account the amplitude scattering formalism, we can write the following relations: 2
L Ipp ¼ jS11 j;
ð13Þ
L 2 jS21 j;
ð14Þ
Ips ¼
2
L Iss ¼ jS22 j;
ð15Þ
L 2 jS12 j;
ð16Þ
Isp ¼
where Iss , Ipp , Ips and Isp are scattered intensities of light, which are physically equivalent to those introduced in Section 3.
5. Orientation of a metal particle in the electrooptical trap Application of TMM theory to our particular case requires some additional information on the behavior of the trapped particle during the experiment. If we want to model the nonspherical particle by the ellipsoid of rotation, we must outline the possible space orientations of the spheroid natural reference frame X 0 Y 0 Z 0 to the laboratory reference frame XYZ, that is to find the Euler angles a and b, being the parameters of the Tmatrix formalism. Y and Z axes of the laboratory reference frame are chosen to lie in the plane perpendicular to the quadrupole electrode axes, and X axis is parallel to the direction of the laser beam, as shown in Fig. 7. As the experiment shows, the trapped droplet performs oscillations along some well-defined direction in the YZ plane of the laboratory reference frame. The particle motion and the particle orientation which cannot be seen by visual observation, are influenced by the electrodynamic force of
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
Fig. 7. Schematic diagram of the electro-optical trap and illustration of the expected orientation of a spheroidal particle in the trap.
the quadrupole and by the light gradient force of the laser beam at proportions depending on distance of the particle from the light beam axis and from the quadrupole axis. The possible particle orientations are due to the direction of the electric field, which acts on the trapped particle in YZ plane of the trap quadrupole. The electrical potentials applied to the quadrupole electrodes are u13 ¼ udc þ uac cosðxtÞ
177
characterized by the angle bel ’ 55° with respect to the Y axis of the laboratory reference frame (see Fig. 8). This corresponds to the angle estimated from the analysis of the CCD camera image (Fig. 8) illustrating the two extreme positions A and B of the particle oscillatory motion. The electric field is expected to affect the spheroidal particle orientation due to the electric dipole induced by that field. The electrical field inside the metallic spheroid should be zero for the entire duration of the particle motion in the trap. To satisfy this condition, the electrical dipole has to be induced on the surface of the particle to compensate the outside field, this means, that the induced electrical dipole of the particle must oscillate along the same direction as the quadrupole field vector, but in opposite phase. The presence of an induced electrical dipole moment in a spheroidal metal particle will orientate its axis of rotation along the direction of the quadrupole trapping field, according to the minimization energy principle. This means that the major axis of a spheroidal particle will be oriented along a fixed direction characterized by angles bel ’ 55° in the YZ plane and finally the following Euler angles: a ¼ 90° and b ¼ bel ’ 55° (see Fig. 7). In the context of the observed depolarization phenomena, the spacial orientation of the spheroidal particle is of much more importance than
ð17Þ
for one pair of electrodes, and u24 ¼ udc uac cosðxtÞ
ð18Þ
for the other one, where udc is a constant voltage applied to all four electrodes, uac is the amplitude of the alternating voltage at frequency x. Dynamic numerical simulation of the electric field magnitude and direction was performed in order to map the time dependence of the electric field vector in the YZ plane of the quadrupole trap for arbitrary values of the parameters udc ; uac and x. For the input parameters udc ¼ 40 V, uac ¼ 338 V and x ¼ 13 190 s1 used in the experiment, the electric field of the quadrupole, acting on the trapped particle during its motion, oscillates in the YZ plane along a fixed direction
Fig. 8. A sample of CCD image of a trapped sodium particle in the electro-optical trap and illustration the effective electric field direction in the trap resulting from the dynamic numerical analysis.
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the dynamic of its mass center. But to obtain more complete picture of particle behavior in the trap let us discuss shortly the kinds of forces acting on the particle mass center as well. The following forces can be identified: the electrodynamic force, the light gradient force, the gravitational force and the damping force. The electrodynamic force was discussed above; it is responsible for particleÕs spacial orientation and for its translational motion. The light gradient force acts on the particleÕs electric dipole moment and can have two possible senses: towards the center of the laser beam or opposite. Under experimental conditions we observe that the particle is pushed out from the laser beam, so we can conclude that the light gradient force acts downwards on the object trapped below the laser beam axis and coincides with the direction of the gravitational force. The damping force
depends on the medium viscosity coefficient, the particle size and the velocity of the particle mass center. The trajectory of the mass center of the trapped particle can be described by solving the Mathieu equation [15].
6. Results To describe scattering of light by a single prolate spheroidal particle we use the TMM Fortran code developed by Mishchenko and Travis [16]. The adjustable pa- rameters of the theory are: the refractive index of the particle, the Euler angels a and b, denning the orientation of the spheroid, the size parameters a and b of the spheroid defining its semi-axis lengths. The equivalent size characterization can be done by pairs of parameters, e.g.,
(a)
(b)
(c)
Fig. 9. The result of numerical simulation of scattered light intensities Iss , Ipp , Ips and Isp at kscat ¼ 647 nm and of the depolarization coefficients dsp and dps as a function of the effective surface radius ref for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and sphericity ratio e ¼ 0:3, 0.5 and 0.67 for (a), (b) and (c), respectively.
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
(e; a(or b)) or (e; ref , where e ¼ a=b is the sphericity ratio and ref is the effective surface radius denning the surface area of a sphere equal to the surface area of a spheroid with a given sphericity ratio e. Simulations are made for a metal particle with the complex refractive index 0:418 þ i2:661 (for k ¼ 647 nm) which is the square root of the Drude dielectric function for metals with the electron damping rate c ¼ 0:54 eV [17], and the bulk plasmon frequency xpl ¼ 5:6 eV. The wavelength of the incident electromagnetic field corresponds to the wavelength of the light field of the probe beam, k ¼ 647 nm. The Euler angels used in simulations corresponds to a ¼ 90° and b ¼ 55°, the values corresponding to the particle orientation in the trap, as discussed in the previous section. Series of scattered intensities Iss , Ipp , Ips and Isp were calculated as a function of the surface effective radius ref
179
of the prolate spheroidal particle of the sphericity ratio e ¼ a=b ¼ 0:3, 0.5 and 0.67 correspondingly and were stored in the data base created for this purpose. To select and to process the data from the data base, a Delphi code was developed and applied. This type of solution saves a lot of processing time in the case of real-time analysis. The developed application can function with different databases and assures high flexibility in solving different simulation problems. The resulting scattered intensities Iss , Ipp , Ips and Isp and the depolarization parameters dsp and dps are presented in Fig. 9. Using a similar method, several series of scattered intensities Iss , Ipp , Ips and Isp as a function of the sphericity ratio e for the surface effective radius ref ¼ 200, 300 and 500 nm were calculated and are illustrated in Fig. 10 together with the corresponding depolarization parameters dps and
(a)
(b)
(c)
Fig. 10. The result of the numerical simulation of scattered light intensities Iss , Ipp , Ips and Isp at kscat ¼ 647 nm and of the depolarization coefficients dsp and dps as a function of the sphericity ratio e for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and surface effective radius ref ¼ 200, 300 and 500 nm for (a), (b) and (c), respectively.
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W. Bazhan et al. / Optics Communications 211 (2002) 171–181
dsp . The limitation in the range of effective radii is due to various problems concerning the convergence of the T-matrix method for particles with different sizes, and sphericity ratios. Usually, at a given wavelength and for a given refractive index, large sizes and small sphericity ratios cause problems. The program execution is automatically terminated if dimensions of certain arrays are not big enough or if the convergence procedure decides that the accuracy of doubleprecision variables is insufficient to obtain a converged T-matrix solution for a given particle. Figs. 9 and 10 show, that effective depolarization takes place when prolate spheroidal particles posses effective radii smaller than the scattered light wavelength, and the larger
the asphericity, the stronger the depolarization effect, as expected. The degree of depolarization is a very sensitive function of the Euler angles describing the spatial orientation of the spheroid. The dependences of the depolarization coefficients dsp and dps on the Euler angles a and b for a particle with surface effective radius ref ¼ 200 nm and the sphericity ratio e ¼ 0:5 is illustrated in Figs. 11 and 12 in the form of three-dimensional maps. If the Euler angles are a ¼ 0° and 90° or b ¼ 0° and b ¼ 90° (so called trivial angle combinations), there is no depolarization, as has been shown by Asano [18,19]. But for some other particle orientations, the degree of depolarization can reach large values due
Fig. 11. Three-dimensional map illustrating the dependence of the depolarization coefficient dps at kscat ¼ 647 nm on the Euler angles a and b for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and effective radius ref ¼ 200 nm.
Fig. 12. Three-dimensional map illustrating the dependence of the depolarization coefficient dsp at kscat ¼ 647 nm on the Euler angles a and b for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and effective radius ref ¼ 200 nm.
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
to a small change in one of the Euler angles. As seen in Figs. 11 and 12, the depolarization coefficients dps and dsp can reach values larger than 50% for parameters concerning the particle size and orientation.
7. Conclusions Relatively large intensities of light scattered by a single levitated particle for ÔcrossedÕ ps and sp polarization geometries were observed. Mie theory, which is formulated for the electromagnetic wave scattered by a sphere, is inadequate to describe this phenomenon. The TMM formalism for spheroidal particles was used for qualitative interpretation of the observed ÕcrossedÕ intensities Ips and Isp . It was shown, that the large depolarization coefficients dps and dsp can be reached for certain spacial orientations of the scatterer with an effective radius smaller than the scattered light wavelength. For the trivial Euler angle combinations a ¼ 0° and 90° or b ¼ 0° and b ¼ 0° and b ¼ 90°, there is no depolarization by the spheroidal particle. The same is true for any combination of the Euler angles, if the sphericity ratio e is equal to 1.0 (the case of a sphere). The scattered intensities and the depolarization coefficients are sensitive functions of the size parameters a and b of the spheroid, and of its orientation. The analysis of trapped particle behavior in the electro-optical trap resulted in the Euler angles a and b describing the trapped particle orientation. Combined with the TMM formalism for spheroidal particles, the depolarization phenomena observed in the experiment can only be explained qualitatively. We have no direct experimental evidence, that the trapped particle is of spheroidal shape, because its size is comparable to the light wavelength and therefore its image gives no information on the shape. The qualitative comparison could be made only for assumed shape and size parameters of the scattering particle of the known index of refraction. The spheroidal shape assumed for numerical analysis was the simplest
181
shape geometry explaining the large depolarization coefficients observed.
Acknowledgements The authors would like to thank Dr M.I. Mishchenko for making available the Fortran code and other sources very helpful in the performed numerical simulations. This work was partially supported by the Polish State Committee for Scientific Research (KBN), Grant No. 2 P03B 102 22.
References [1] M. Born, E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1970. [2] P.C. Waterman, Phys. Rev. D 3 (1971) 825. [3] M.I. Mishchenko, L.D. Travis, D.W. Mackowski, J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 535. [4] W. Paul, Rev. Mod. Phys. 62 (1990) 531. [5] K. Sasaki, M. Tsukima, H. Masuhara, Appl. Phys. Lett. 71 (1997) 37. [6] P. Markowicz, D. Jakubczyk, K. Kolwas, M. Kolwas, J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 3605. [7] P. Markowicz, D. Jakubczyk, K. Kolwas, M. Kolwas, J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 5513. [8] C.R. Vidal, J. Cooper, J. Appl. Phys. 40 (1969) 3370. [9] K. Kolwas, M. Kolwas, D. Jakubczyk, Appl. Phys. B 60 (1995) 173. [10] S. Demianiuk, K. Kolwas, Appl. Phys. B 60 (2001) 1651. [11] L. Tsang, J.A. Kong, R.T. Shin, Theory of Microwave Remote Sensing, Wiley, New York, 1985. [12] M.I. Mishchenko, J.W. Hovenier, L.D. Travis, Light Scattering by Non-spherical Particles, Academic Press, SanDiego, 2000. [13] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988. [14] M.I. Mishchenko, Appl. Opt. 39 (2000) 1026. [15] R.F. Wuerker, H. Shelton, R.V. Langmuir, J. Appl. Phys. 30 (1959) 342. [16] M.I. Mishchenko, L.D. Travis, J. Quant. Spectrosc. Radiat. Transfer 60 (2000) 309. [17] P. Markowicz, K. Kolwas, M. Kolwas, Phys. Lett. A 236 (1997) 543. [18] S. Asano, G. Yamamoto, Appl. Opt. 14 (1975) 29. [19] S. Asano, M. Sato, Appl. Opt. 19 (1980) 962.
Optics Communications 211 (2002) 171–181 www.elsevier.com/locate/optcom
Depolarization of light scattered by a single sodium nanoparticle trapped in an electro-optical trap W. Bazhan *, K. Kolwas, M. Kolwas Institute of Physics of Polish Academy of Sciences, Al. Lotnik ow 32/46, 02-668 Warsaw, Poland Received 8 April 2002; received in revised form 21 June 2002; accepted 7 August 2002
Abstract Scattering of polarized light by a single metal particle is studied. Single sodium particles trapped in an electro-optical trap show the unexpectedly large cross-polarized scattered intensities not predicted by Mie theory. To explain qualitatively the large depolarization effect, spheroidal shape of the particle is assumed and the T-matrix method (TMM) is applied to study the corresponding scattered intensities and the resulting depolarization rates. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 41.20.)q; 42.25.Ja; 42.68.Mj; 77.22.Ej Keywords: Particle trapping; Scattering of light by nonspherical objects; T-matrix method
1. Introduction Quantitative interpretation of light scattering by small particles (aerosols, clouds, colloidal particles, large metal clusters and so on) requires accurate knowledge of the interaction of small objects with electromagnetic radiation. Although many natural particles are known to have nonspherical shapes, the relative simplicity and easy applicability of Mie theory [1] has led to widespread practice of treating scattering particles as spheres. In fact, a vast amount of accumulated evidence suggests that scattering properties of
*
Corresponding author. Fax: +48-22-843-0926. E-mail address: [email protected] (W. Bazhan).
nonspherical particles can differ both quantitatively and qualitatively from those of ÔequivalentÕ spheres. One of the most popular modern techniques of modeling the light scattering by nonspherical objects is the T-matrix method (TMM) firstly introduced by Waterman [2] and significantly improved by Mishchenko [3]. Most papers published during the last 20 years, employed the TMM for interpreting the light scattering by randomly oriented axially symmetric mono- or polydisperse nonabsorbing particles. But very few of them were dedicated to applying the T-matrix techniques to the case of light scattering by single, free particles and especially by single free metal particles. Such tiny objects have rarely been examined under laboratory conditions, what has mainly been caused by difficulties in fabrication
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 9 9 - 0
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and localization for investigation. Currently the most developed nano- and microparticle localizing techniques is particle trapping. There are many reports concerning various combinations of particle trap construction (e.g. [4,5]) and trapping conditions. But the area of metallic particle trapping in conducting gaseous media is still virtually unexplored since it presents some substantial technical difficulties [6,7]. In this paper we report the results of experimental investigation of scattering of linearly polarized laser light by single trapped sodium clusters in a cold plasma environment. We used the TMM tools for interpreting the unexpectedly large depolarization of the scattered light observed in our experiments.
2. Experimental setup Sodium clusters are produced in a heat-pipe type stainless-steel cell [8] with crossed tube design [9,10]. The inner part of the cell is heated by an electrical oven, which ensures a uniform central temperature region. In standard working conditions used in the present experiments, the temperature is stabilized at 650 K. This temperature ensures an appropriate sodium vapor density, essential for a sufficient amount of Na2 molecules (dimers) over the sodium deposit placed in the central part of the cell (see [10]). The presence of dimers enables to supersaturate the vapor by light, the process described in detail in earlier papers [9,10]. The arms of the cell are cooled with flowing water in order to keep glass cell windows at a safe temperature. To prevent sodium deposition on the cooled windows of the cell, we buffer it with helium gas. Due to the heat-pipe features of the cell [8], sodium contaminants which could not be avoided during preparation of the sodium deposit, are pushed out from the central region of the cell towards colder arms, owing to the high sodium vapor pressure in the central high temperature region. Helium buffer gas, which is chemically neutral with respect to sodium, plays two further important roles due to its high pressure (in the present experiment typically about 640 Torr). In the
process of collisions with dimers, helium atoms effectively assist the process of dissociation of dimers previously excited by laser light. In addition, the helium gas serves as a reservoir of heat released during cluster formation, enabling effective spontaneous growth of clusters in supersaturated sodium vapor. Two types of clusters are produced in the cell. The first type are Ôlight inducedÕ in the central, hot region of the cell. The hot clusters, sodium droplets, appear only in the presence of the laser light with frequency in resonance with the well-defined sodium dimer transition enabling excitation of an excess of atoms in previously saturated vapor, constituting the source of supersaturation and enabling the spontaneous growth of sodium droplets from atomic size scale up to large sodium droplets of about 300 nm in diameter (see [10] for details). The process of sodium cluster growth lasts from few to few tens of seconds after laser light with sufficiently large power (100–400 mW) is switched on. Clusters of a second type are produced in the arms of the cell, in the gradient temperature zones near the windows. Sodium vapor reaching that region is cooled down and thus supersaturated, becoming the source of the large sodium clusters growing on contaminations (oxides and hydrates of sodium) collected in those zones. Such contaminants constitute condensation embryos leading to clusters whose shapes are expected to differ in an important manner from the spherical sodium droplets induced by light. Even at relatively small oven temperatures, and low laser light power, we can see a lot of particles localized in cold parts of the cell arms. To investigate a single particle by optical means we use an electro-optical trap described in more detail in [6,7]. Its electric part consists of the quadrupole [4] allowing for trapping in two dimensions, adapted for working in a hot dense gaseous medium. The quadrupole consists of four (cylindrical) metal rods which are 1 mm apart and are electrically insulated with fused-silica tubes (see Fig. 1). This insulation is essential for trapping particles the electrodes. On the other hand the presence of the conducting medium greatly reduces accumulation of charged clusters on the electrode
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
Fig. 1. Schematic diagram of the electro-optical trap.
insulation, so that there is less distortion in the shape of the trapping electric potential. The trapping in the third dimension is assured by the light gradient force [7] of the 488 nm line of the Arþ laser working at 400 mW. The trapping beam is directed perpendicular to the quadrupole axis. The trap is mounted in the central region of the cell. The experimental setup is sketched in Fig. 2. A detailed discussion of the trapping mechanism and the description of the papers [6,7]. Our previous experiments were directed to trap the sodium droplets generated by light. However, we noticed, that after the trap is switched on, the Ôcold armsÕ clusters were efficiently driven by the electrostatic forces into the trap, in which they could be kept for a long time (some tens of minutes). In contrast to the light induced sodium droplets, those large particles are expected to be of irregular shape. Scattering experiments performed
Fig. 2. Schematic diagram of the experimental setup.
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on those objects produced such rarely seen result (up to 100% of light depolarization in the direction of observation), that it itself became a subject of investigation. The scattering experiment on the trapped particle is performed with the additional (40 mW) probe light beam of the Argon–Krypton laser at wavelength of 647 nm (with the aperture of about 5 mm), directed perpendicular to the quadrupole axis and opposite to the Arþ trapping laser beam. In contrary to the Arþ laser beam, the probe beam is not focused. In such conditions, the stability of the particle inside the trap is not disturbed by the presence of the weak probe beam and the probe intensity which illuminates the particle is almost isotropic. The intensity of the elastically scattered light from the probe beam is observed at right angle to the direction of the probe beam, and after passing through a polarization analyzer, a color filter and a microscope system, is measured by a monochromatic CCD camera installed on the microscope eyepiece (see Fig. 2).
3. Experimental scattered intensities The intensity of elastically scattered probe beam light is measured as a function of time at right angle to the direction of the laser beam propagation. The experimental signal is gathered in four different polarization geometries: (i) The polarization of the incident light perpendicular to the observation plane, and the polarization of the observed light perpendicular or parallel to the observation plane (on the basis of the usual s–p convention, we denote this as ss and sp polarization geometry, respectively (see Fig. 3)). (ii) The polarization of the incident light parallel to the observation plane and the polarization of observed light parallel or perpendicular to the observation plane (pp and ps polarization geometry, respectively (see Fig. 3)). The positions of the polarizer and of the analyzer assuring respective sequential combinations: ss ! sp ! pp ! ps are controlled by a PC with appropriate measurement of the time intervals. The complete time of a single measurement is
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Fig. 3. Polarization geometries of the scattering experiment.
nearly 5 s and the time between two sequential measurements adds to a total of 10 s. Electronic control of the relative positions of polarizers, of the data acquisition and numerical processing of the scattered intensities in the CCD camera image allowed an improvement in the quality of results with respect to our previous trapping experiment [6,7]. Figs. 4 and 5 illustrate the measured intensities of the light scattered by a single cluster during its residence in the trap and the resulting depolarization coefficients dps and dsp : dps ¼ 100 Ips =Ipp ;
ð1Þ
dsp ¼ 100 Isp =Iss :
ð2Þ
Fig. 4. The intensities Iss , Ipp (ÔparallelÕ polarization geometries) and Ips , Isp (ÔcrossedÕ polarization geometries) of light scattered by a single trapped sodium particle, measured during the residence of the particle in the trap for kscat ¼ 647 nm, uac ¼ 338 V, x ¼ 13 190 s1 and udc ¼ 40 V.
Fig. 5. The depolarization coefficients dps and dsp for light scattered by a single trapped sodium particle, measured during the residence of particle in the trap for kscat ¼ 647 nm, uac ¼ 338 V, x ¼ 13 190 s1 and udc ¼ 40 V.
A single experimental measurement comprises of 16 serial images for the four sequential combinations of the polarizer–analyzer system: ss ! sp ! pp ! ps. The observed time dependencies of the intensities Iss and Ipp of the scattered fight for ss and pp polarizations, respectively (see Fig. 4), show that the trapped cluster changes its size with time. As illustrated in Figs. 4 and 5, we observed also relatively large intensities Ips and Isp for the ÔcrossedÕ ps and sp polarizations. Mie scattering theory does not predict non-zero intensity corresponding to the ÔcrossedÕ polarization geometries: the depolarized components Ips and Isp in the scattered light are absent. Mie theory is formulated for the electromagnetic wave scattered by a sphere with a smooth surface and sharp boundary and with a well-defined dielectric function, which is an adjustable parameter of the theory. It relies on solving MaxwellÕs equations together with the boundary conditions separated into a set of ordinary differential equations for the two subfields (internal and external) in the form of infinite series [1]. We interpret the presence of the cross-polarization intensity components in scattering experiments as being due to the fact that the investigated scatterers are not perfect spherical objects. If so, Mie theory cannot be used for adequate interpretation of the observed depolarization of the scattered light.
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The aim of the present paper is to check, whether the hypothesis concerning the nonsphericity of the shape of the scatterer is sufficient to interpret the presence of unexpectedly large crosspolarization intensities Ips and Isp of the ÔcrossedÕ ps and sp polarization geometries. For this purpose we used the formalism of the TMM for spheroidal particles as relatively the most simple shapes of scatterers one can assume for explaining the observations. The appropriate computer code created for that purpose results in qualitative agrement with experiment that cannot be achieved by means of Mie theory.
4. TMM scattering intensities The main purpose of this section is to recall the theoretical basis of the TMM and to use it for interpretation of the scattered light depolarization effects observed in our experiments. TMM, or Watermans extended boundary condition method [2] (EBCM), has shown its high potential for studying scattering by nonspherical particles. Although it can be applied to virtually any kind of nonspherical particles, it is most efficient for axisymmetric scatterers. A detailed review of the EBCM is given by Tsang et al. [11] and Mishchenko et al. [12]. In the T-matrix approach, the incident Einc and the scattered Esca fields are expanded into series of vector spherical harmonics Mmn and Nmn , respectively, as follows: inc
E ðRÞ ¼
1 X n X
½amn RgMmn ðkRÞ
n¼1 m¼n
þ bmn RgNmn ðkRÞ ; Esca ðRÞ ¼
1 X
n X
ð3Þ
½pmn Mmn ðkRÞ
n¼1 m¼n
þ qmn Nmn ðkRÞ ;
jRj > r0 ;
ð4Þ
where k ¼ 2p=k is the free-space wavenumber for free-space wavelength k; R is the radius vector with its origin at the origin of the coordinate system; r0 is the radius of a circumscribing sphere of the scattering particle; amn ; bmn and pmn ; qmn are incident field expansion coefficients and scattered field
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expansion coefficients appropriately. The analytical formulas for amn and bmn are introduced by Waterman [2]. The expressions for the functions RgMmn and RgNmn can be obtained from the functions Mmn and Nmn by replacing the spherical Hankel functions hnð1Þ in its definition by the spherical Bessel functions jn as it was shown by Mishchenko et al. [12]. In addition, the internal field is also expanded as follows: Eint ðRÞ ¼
1 X n X
½cmn RgMmn ðmr kRÞ
n¼1 m¼n
þ dmn RgNmn ðmr kRÞ ;
ð5Þ
where mr is the refractive index of the particle relative to that of the surrounding medium; cmn and qmn are internal field expansion coefficients. The relation between the scattered field coefficients pmn and qmn and the incident field coefficients amn and bmn is linear and is given by a transition matrix T [2,11]: pmn ¼
1 n0 X X
11 12 Tmnm 0 n0 am0 n0 þ Tmnm0 n0 bm0 n0 ;
ð6Þ
n0 ¼1 m0 ¼n
qmn ¼
1 n0 X X
21 22 Tmnm 0 n0 am0 n0 þ Tmnm0 n0 bm0 n0 :
ð7Þ
n0 ¼1 m0 ¼n
In the more compact matrix notation, Eqs. (6) and (7) can be written as 11 P a T T12 a ¼T ¼ : ð8Þ q b T21 T22 b Similarly, the relation between the exciting field coefficients and the internal field coefficients is 11 a Q12 c Q ; ð9Þ ¼ b Q21 Q22 d where the elements of the matrix Q are twodimensional integrals, which must be evaluated numerically over the particle surface, and depend on the particle size, shape, refractive index, and orientation with respect to the natural reference frame [11]. As the scattered field coefficients can also be expressed by the internal field coefficients p RgQ11 RgQ12 c ¼ ; ð10Þ q RgQ21 RgQ22 d
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The T matrix can be represented as 1
T ¼ RgQ½Q :
ð11Þ
The scattering properties of a particle are completely characterized by the elements of the amplitude scattering matrix that relates the scattered field to the the incident filed. The amplitude scattering matrix can be determining as sca inc expðikRÞ L sca inc Ep Ep S ðn ; n ; a; b; cÞ ¼ inc ; Esca R E s s ð12Þ L
where S is the 2 2 amplitude matrix that transforms the electric field vector components Epinc ; Esinc of the incident wave into the electric field vector components Epsca ; Essca of the scattered wave in the laboratory reference frame. The latter is a right-handed Cartesian coordinate system, with orientation fixed in space, having its origin inside the particle [12]. The direction of propagation of a transverse electromagnetic wave is specified by a unit vector n or, equivalently, by a pair (hL ; uL ) of polar and azimuthal angles (see Fig. 6). The amplitude matrix depends on the directions of the incident and of the observed scattered light as well as on the size, morphology, and the particle natural reference frame with respect to the laboratory
z n φ θ θ y φL
x Fig. 6. The laboratory coordinate system.
reference frame as specified by the Euler angles of rotation a, b, and c [13]. In order to calculate the SL amplitude matrix, the T matrix should first be found for a given scattering particle. It is rather convenient to specify the amplitude matrix with respect to the particle reference frame by using the T-matrix method, and then to obtain the SL amplitude matrix by means of the coordinate system transformation, as has been explicitly shown by Mishchenko [14]. By taking into account the amplitude scattering formalism, we can write the following relations: 2
L Ipp ¼ jS11 j;
ð13Þ
L 2 jS21 j;
ð14Þ
Ips ¼
2
L Iss ¼ jS22 j;
ð15Þ
L 2 jS12 j;
ð16Þ
Isp ¼
where Iss , Ipp , Ips and Isp are scattered intensities of light, which are physically equivalent to those introduced in Section 3.
5. Orientation of a metal particle in the electrooptical trap Application of TMM theory to our particular case requires some additional information on the behavior of the trapped particle during the experiment. If we want to model the nonspherical particle by the ellipsoid of rotation, we must outline the possible space orientations of the spheroid natural reference frame X 0 Y 0 Z 0 to the laboratory reference frame XYZ, that is to find the Euler angles a and b, being the parameters of the Tmatrix formalism. Y and Z axes of the laboratory reference frame are chosen to lie in the plane perpendicular to the quadrupole electrode axes, and X axis is parallel to the direction of the laser beam, as shown in Fig. 7. As the experiment shows, the trapped droplet performs oscillations along some well-defined direction in the YZ plane of the laboratory reference frame. The particle motion and the particle orientation which cannot be seen by visual observation, are influenced by the electrodynamic force of
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Fig. 7. Schematic diagram of the electro-optical trap and illustration of the expected orientation of a spheroidal particle in the trap.
the quadrupole and by the light gradient force of the laser beam at proportions depending on distance of the particle from the light beam axis and from the quadrupole axis. The possible particle orientations are due to the direction of the electric field, which acts on the trapped particle in YZ plane of the trap quadrupole. The electrical potentials applied to the quadrupole electrodes are u13 ¼ udc þ uac cosðxtÞ
177
characterized by the angle bel ’ 55° with respect to the Y axis of the laboratory reference frame (see Fig. 8). This corresponds to the angle estimated from the analysis of the CCD camera image (Fig. 8) illustrating the two extreme positions A and B of the particle oscillatory motion. The electric field is expected to affect the spheroidal particle orientation due to the electric dipole induced by that field. The electrical field inside the metallic spheroid should be zero for the entire duration of the particle motion in the trap. To satisfy this condition, the electrical dipole has to be induced on the surface of the particle to compensate the outside field, this means, that the induced electrical dipole of the particle must oscillate along the same direction as the quadrupole field vector, but in opposite phase. The presence of an induced electrical dipole moment in a spheroidal metal particle will orientate its axis of rotation along the direction of the quadrupole trapping field, according to the minimization energy principle. This means that the major axis of a spheroidal particle will be oriented along a fixed direction characterized by angles bel ’ 55° in the YZ plane and finally the following Euler angles: a ¼ 90° and b ¼ bel ’ 55° (see Fig. 7). In the context of the observed depolarization phenomena, the spacial orientation of the spheroidal particle is of much more importance than
ð17Þ
for one pair of electrodes, and u24 ¼ udc uac cosðxtÞ
ð18Þ
for the other one, where udc is a constant voltage applied to all four electrodes, uac is the amplitude of the alternating voltage at frequency x. Dynamic numerical simulation of the electric field magnitude and direction was performed in order to map the time dependence of the electric field vector in the YZ plane of the quadrupole trap for arbitrary values of the parameters udc ; uac and x. For the input parameters udc ¼ 40 V, uac ¼ 338 V and x ¼ 13 190 s1 used in the experiment, the electric field of the quadrupole, acting on the trapped particle during its motion, oscillates in the YZ plane along a fixed direction
Fig. 8. A sample of CCD image of a trapped sodium particle in the electro-optical trap and illustration the effective electric field direction in the trap resulting from the dynamic numerical analysis.
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the dynamic of its mass center. But to obtain more complete picture of particle behavior in the trap let us discuss shortly the kinds of forces acting on the particle mass center as well. The following forces can be identified: the electrodynamic force, the light gradient force, the gravitational force and the damping force. The electrodynamic force was discussed above; it is responsible for particleÕs spacial orientation and for its translational motion. The light gradient force acts on the particleÕs electric dipole moment and can have two possible senses: towards the center of the laser beam or opposite. Under experimental conditions we observe that the particle is pushed out from the laser beam, so we can conclude that the light gradient force acts downwards on the object trapped below the laser beam axis and coincides with the direction of the gravitational force. The damping force
depends on the medium viscosity coefficient, the particle size and the velocity of the particle mass center. The trajectory of the mass center of the trapped particle can be described by solving the Mathieu equation [15].
6. Results To describe scattering of light by a single prolate spheroidal particle we use the TMM Fortran code developed by Mishchenko and Travis [16]. The adjustable pa- rameters of the theory are: the refractive index of the particle, the Euler angels a and b, denning the orientation of the spheroid, the size parameters a and b of the spheroid defining its semi-axis lengths. The equivalent size characterization can be done by pairs of parameters, e.g.,
(a)
(b)
(c)
Fig. 9. The result of numerical simulation of scattered light intensities Iss , Ipp , Ips and Isp at kscat ¼ 647 nm and of the depolarization coefficients dsp and dps as a function of the effective surface radius ref for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and sphericity ratio e ¼ 0:3, 0.5 and 0.67 for (a), (b) and (c), respectively.
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(e; a(or b)) or (e; ref , where e ¼ a=b is the sphericity ratio and ref is the effective surface radius denning the surface area of a sphere equal to the surface area of a spheroid with a given sphericity ratio e. Simulations are made for a metal particle with the complex refractive index 0:418 þ i2:661 (for k ¼ 647 nm) which is the square root of the Drude dielectric function for metals with the electron damping rate c ¼ 0:54 eV [17], and the bulk plasmon frequency xpl ¼ 5:6 eV. The wavelength of the incident electromagnetic field corresponds to the wavelength of the light field of the probe beam, k ¼ 647 nm. The Euler angels used in simulations corresponds to a ¼ 90° and b ¼ 55°, the values corresponding to the particle orientation in the trap, as discussed in the previous section. Series of scattered intensities Iss , Ipp , Ips and Isp were calculated as a function of the surface effective radius ref
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of the prolate spheroidal particle of the sphericity ratio e ¼ a=b ¼ 0:3, 0.5 and 0.67 correspondingly and were stored in the data base created for this purpose. To select and to process the data from the data base, a Delphi code was developed and applied. This type of solution saves a lot of processing time in the case of real-time analysis. The developed application can function with different databases and assures high flexibility in solving different simulation problems. The resulting scattered intensities Iss , Ipp , Ips and Isp and the depolarization parameters dsp and dps are presented in Fig. 9. Using a similar method, several series of scattered intensities Iss , Ipp , Ips and Isp as a function of the sphericity ratio e for the surface effective radius ref ¼ 200, 300 and 500 nm were calculated and are illustrated in Fig. 10 together with the corresponding depolarization parameters dps and
(a)
(b)
(c)
Fig. 10. The result of the numerical simulation of scattered light intensities Iss , Ipp , Ips and Isp at kscat ¼ 647 nm and of the depolarization coefficients dsp and dps as a function of the sphericity ratio e for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and surface effective radius ref ¼ 200, 300 and 500 nm for (a), (b) and (c), respectively.
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dsp . The limitation in the range of effective radii is due to various problems concerning the convergence of the T-matrix method for particles with different sizes, and sphericity ratios. Usually, at a given wavelength and for a given refractive index, large sizes and small sphericity ratios cause problems. The program execution is automatically terminated if dimensions of certain arrays are not big enough or if the convergence procedure decides that the accuracy of doubleprecision variables is insufficient to obtain a converged T-matrix solution for a given particle. Figs. 9 and 10 show, that effective depolarization takes place when prolate spheroidal particles posses effective radii smaller than the scattered light wavelength, and the larger
the asphericity, the stronger the depolarization effect, as expected. The degree of depolarization is a very sensitive function of the Euler angles describing the spatial orientation of the spheroid. The dependences of the depolarization coefficients dsp and dps on the Euler angles a and b for a particle with surface effective radius ref ¼ 200 nm and the sphericity ratio e ¼ 0:5 is illustrated in Figs. 11 and 12 in the form of three-dimensional maps. If the Euler angles are a ¼ 0° and 90° or b ¼ 0° and b ¼ 90° (so called trivial angle combinations), there is no depolarization, as has been shown by Asano [18,19]. But for some other particle orientations, the degree of depolarization can reach large values due
Fig. 11. Three-dimensional map illustrating the dependence of the depolarization coefficient dps at kscat ¼ 647 nm on the Euler angles a and b for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and effective radius ref ¼ 200 nm.
Fig. 12. Three-dimensional map illustrating the dependence of the depolarization coefficient dsp at kscat ¼ 647 nm on the Euler angles a and b for a prolate spheroidal particle with refractive index 0:418 þ i2:661 and effective radius ref ¼ 200 nm.
W. Bazhan et al. / Optics Communications 211 (2002) 171–181
to a small change in one of the Euler angles. As seen in Figs. 11 and 12, the depolarization coefficients dps and dsp can reach values larger than 50% for parameters concerning the particle size and orientation.
7. Conclusions Relatively large intensities of light scattered by a single levitated particle for ÔcrossedÕ ps and sp polarization geometries were observed. Mie theory, which is formulated for the electromagnetic wave scattered by a sphere, is inadequate to describe this phenomenon. The TMM formalism for spheroidal particles was used for qualitative interpretation of the observed ÕcrossedÕ intensities Ips and Isp . It was shown, that the large depolarization coefficients dps and dsp can be reached for certain spacial orientations of the scatterer with an effective radius smaller than the scattered light wavelength. For the trivial Euler angle combinations a ¼ 0° and 90° or b ¼ 0° and b ¼ 0° and b ¼ 90°, there is no depolarization by the spheroidal particle. The same is true for any combination of the Euler angles, if the sphericity ratio e is equal to 1.0 (the case of a sphere). The scattered intensities and the depolarization coefficients are sensitive functions of the size parameters a and b of the spheroid, and of its orientation. The analysis of trapped particle behavior in the electro-optical trap resulted in the Euler angles a and b describing the trapped particle orientation. Combined with the TMM formalism for spheroidal particles, the depolarization phenomena observed in the experiment can only be explained qualitatively. We have no direct experimental evidence, that the trapped particle is of spheroidal shape, because its size is comparable to the light wavelength and therefore its image gives no information on the shape. The qualitative comparison could be made only for assumed shape and size parameters of the scattering particle of the known index of refraction. The spheroidal shape assumed for numerical analysis was the simplest
181
shape geometry explaining the large depolarization coefficients observed.
Acknowledgements The authors would like to thank Dr M.I. Mishchenko for making available the Fortran code and other sources very helpful in the performed numerical simulations. This work was partially supported by the Polish State Committee for Scientific Research (KBN), Grant No. 2 P03B 102 22.
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